Optimal. Leaf size=69 \[ \frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} d} \]
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Rubi [A] time = 0.109976, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2676, 2667, 63, 206} \[ \frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} d} \]
Antiderivative was successfully verified.
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Rule 2676
Rule 2667
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{1}{2} a^2 \int \sec (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{d}\\ &=-\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{2} d}+\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}\\ \end{align*}
Mathematica [A] time = 0.368171, size = 75, normalized size = 1.09 \[ \frac{a^2 \left (-\frac{2 \sqrt{a (\sin (c+d x)+1)}}{\sin (c+d x)-1}-\sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 66, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{d} \left ({\frac{1}{a\sin \left ( dx+c \right ) -a}\sqrt{a+a\sin \left ( dx+c \right ) }}+{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+a\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75117, size = 263, normalized size = 3.81 \begin{align*} \frac{\sqrt{2}{\left (a^{2} \sin \left (d x + c\right ) - a^{2}\right )} \sqrt{a} \log \left (-\frac{a \sin \left (d x + c\right ) - 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}}{4 \,{\left (d \sin \left (d x + c\right ) - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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